Acceleration

Acceleration is a key concept in physics that alludes to the rate at which an object's velocity changes over time. As a vector quantity, it possesses both magnitude and direction. Acceleration takes place when a body  speeds up, slows down, or alters its direction. We experience acceleration in many everyday scenarios, such as a car accelerating from a stoplight, a roller coaster picking up speed on a descent, or a planet orbiting the sun. Grasping the concept of acceleration is essential not only for physics but also for applications in engineering, sports science, and medicine.

1.0Acceleration

• Rate of change of velocity is called acceleration.
• It is a vector quantity.
• Its direction is the same as that of change in velocity (not in the direction of the velocity).
• Dimension Formula: [M⁰L¹T ^–2]
• Unit : (S.I.)  m/s²   (C.G.S.) cm/s²
• There are 3 ways to change a velocity (vector)
• Only magnitude change      
• Only direction change          
• Both direction or magnitude change

2.0Examples Of Acceleration

• Gravity: When an object falls, like an apple dropping from a tree, it accelerates downward toward the ground because of the force of gravity.
• Driving a Car: When you press the gas pedal to accelerate from a stop, the car moves forward more quickly. In contrast, applying the brakes causes the car to decelerate, effectively accelerating in the opposite direction.

3.0Types of Acceleration

1. Uniform Acceleration: A body is considered to have uniform acceleration if both the magnitude and direction of its acceleration remain constant throughout the motion of the particle.
2. Non-Uniform Acceleration: A body is described as having non-uniform acceleration if either the magnitude, the direction, or both change during its motion.
3. Average Acceleration: It is the ratio of the total change in velocity to the total time taken by the particle.

${\overrightarrow{\mathrm{a}}}_{\mathrm{avg}}=\frac{\text{ Change in Velocity }}{\text{ Time Interval }}=\frac{\Delta \overrightarrow{\mathrm{v}}}{\Delta \mathrm{t}}=\frac{{\overrightarrow{\mathrm{v}}}_{\mathrm{f}}-{\overrightarrow{\mathrm{v}}}_{\mathrm{i}}}{\Delta \mathrm{t}}$          

Direction of average acceleration is along the change in velocity.

4. Instantaneous Acceleration: Acceleration of an object at a specific moment in time.

${\overrightarrow{\mathrm{a}}}_{\text{inst }}={\lim }_{\Delta \mathrm{t}\to 0}\left(\frac{\overrightarrow{\Delta \mathrm{v}}}{\Delta \mathrm{t}}\right)\Rightarrow {\overrightarrow{\mathrm{a}}}_{\mathrm{inst}}=\frac{\overrightarrow{\mathrm{dv}}}{\mathrm{dt}}$, where $\vec{v}$ is function of time

• First derivative of velocity is called Instantaneous Acceleration

$a=\frac{dv}{dt}=\frac{d^{2}r}{d{t}^2}\left(v=\frac{dr}{dt}\right)$

• Second derivative of position vector is called Instantaneous acceleration

$a=\frac{dv}{dt}\times \frac{dx}{dx}\Rightarrow v\frac{dv}{dx}$, where v is the function of position x

Example: The relationship between time t and distance x  is given by the equation $t=a{x}^2+bx$, where a and b are constants. How can we express the instantaneous acceleration in terms of the instantaneous velocity?

Solution:

$t=a{x}^2+bx$

$\frac{dt}{dx}=2ax+b$

velocity,

$v=\frac{dx}{dt}=(2ax+b{)}^{-1}$

Acceleration,

$a=\frac{dv}{dt}=\frac{dv}{dx}\cdot \frac{dx}{dt}=v\frac{dv}{dx}$

$a=v\frac{d}{dx}(2ax+b{)}^{-1}$

$=v(-1)(2ax+b{)}^{-2}\cdot 2a=-2a(2ax+b{)}^{-3}=-2a{v}^3$

4.0Acceleration Due To Gravity

• When a body falls freely towards the surface of earth, its velocity continuously increases. The acceleration yielded in a freely falling body under the gravitational pull of the Earth is called acceleration due to gravity.
• It is a vector having direction towards the centre of Earth. It does not depend on the mass, size and shape of the body.
• Value of g varies from place to place on the surface of Earth, it depends on altitude, depth, rotation of the Earth and Shape of the Earth.
• Value of $g=9.8\frac{m}{s^2}$ or $32ft{s}^{-2}$

Relation Between Acceleration due to Gravity(g) and Universal Gravitational Constant(G)

$F=G\frac{Mm}{R^2}$ …………… (1)

$F=mg$ …………… (2)

$mg=G\frac{Mm}{R^2}$

$g=\frac{GM}{R^2}$, value of g is the unconstrained mass, size and shape of the body falling under gravity.

5.0Centripetal Acceleration

• When an object is in uniform circular motion, its speed remains constant; however, its velocity continuously changes due to the shifting direction. As a result, the motion is considered accelerated.
• A body in uniform circular motion experiences an acceleration directed radially inward toward the center of its circular path. This acceleration is known as centripetal (or center-seeking) acceleration.
• $a=\frac{v^2}{r}={\omega }^{2}r$
• The direction of centripetal acceleration continuously changes and is always perpendicular to the velocity vector at every point along the path.

6.0Difference Between Acceleration and Velocity